A123023 a(n) = (n-1)*a(n-2), a(0)=1, a(1)=0.

1, 0, 1, 0, 3, 0, 15, 0, 105, 0, 945, 0, 10395, 0, 135135, 0, 2027025, 0, 34459425, 0, 654729075, 0, 13749310575, 0, 316234143225, 0, 7905853580625, 0, 213458046676875, 0, 6190283353629375, 0, 191898783962510625, 0, 6332659870762850625, 0, 221643095476699771875

Offset: 0

Author

Roger L. Bagula, Sep 24 2006

Keywords

• nonn
• easy

Comments

a(n) is the number of ways of separating n terms into pairs. - Stephen Crowley, Apr 07 2007

a(n) is the n-th moment of the standard normal distribution. - Hal M. Switkay, Nov 06 2019

a(n) is the number of fixed-point free involutions in the symmetric group of degree n. - Nick Krempel, Feb 26 2020

Examples

			From _Gus Wiseman_, Dec 23 2018: (Start)
The a(6) = 15 ways of partitioning {1,2,3,4,5,6} into disjoint pairs:
  {{12}{34}{56}},  {{12}{35}{46}},  {{12}{36}{45}},
  {{13}{24}{56}},  {{13}{25}{46}},  {{13}{26}{45}},
  {{14}{23}{56}},  {{14}{25}{36}},  {{14}{26}{35}},
  {{15}{23}{46}},  {{15}{24}{36}},  {{15}{26}{34}},
  {{16}{23}{45}},  {{16}{24}{35}},  {{16}{25}{34}}.
(End)
		

References

• Richard Bronson, Schaum's Outline of Modern Introductory Differential Equations, MacGraw-Hill, New York, 1973, page 107, solved problem 19.18
• Norbert Wiener, Nonlinear Problems in Random Theory, 1958, Equation 1.31

Links

• G. C. Greubel, Table of n, a(n) for n = 0..799
• Mihai Nica, Terence Tao's central limit theorem, Double Factorials (!!) and the Moment Method, YouTube video (2023).
• Sebastian Volz, Design and Implementation of Efficient Algorithms for Operations on Partitions of Sets, Bachelor Thesis, Saarland Univ. (Germany, 2023). See p. 45.
• Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017-2018, Table 2 on p. 15.

Crossrefs

Cf. A000085, A000110, A000124, A000142, A000587, A001147, A001464, A006129, A079267, A124794, A186021, A295193.

Programs

Magma

a:=[1,0]; [n le 2 select a[n] else (n-2)*Self(n-2): n in [1..30]]; // Marius A. Burtea, Nov 07 2019

Maple

with(combstruct): ZL2 := [S, {S=Set(Cycle(Z, card=2))}, labeled]:
seq(count(ZL2, size=n), n=0..36); # Zerinvary Lajos, Sep 24 2007
a := n -> ifelse(irem(n, 2) = 1, 0, 2^(n/2) * pochhammer(1/2, n/2)):
seq(a(n), n = 0..36); # Peter Luschny, Jan 11 2023

Mathematica

RecurrenceTable[{a[0] == 1, a[1] == 0, a[n] == (n - 1) a[n - 2]}, a[n], {n, 0, 31}] (* Ray Chandler, Jul 30 2015 *)

Formula

a(n) = (1/2)*Gamma((1/2)*n + 1/2)*2^((1/2)*n)*(1 + (-1)^n)/sqrt(Pi). - Stephen Crowley, Apr 07 2007

E.g.f.: exp(x^2/2). - Geoffrey Critzer, Mar 15 2009

a(2n) = A001147(n). - R. J. Mathar, Oct 11 2011

From Sergei N. Gladkovskii, Nov 18 2012, Dec 05 2012, May 16 2013, May 24 2013, Jun 07 2013: (Start)

Continued fractions:

E.g.f.: E(0) where E(k) = 1 + x^2*(4*k+1)/((4*k+2)*(4*k+3) - x^2*(4*k+2)*(4*k+3)^2/(x^2*(4*k+3) + (4*k+4)*(4*k+5)/E(k+1))).

G.f.: 1/G(0) where G(k) = 1 - x^2*(k+1)/G(k+1).

G.f.: 1 + x^2/(1+x) + Q(0)*x^3/(1+x), where Q(k) = 1 + (2*k+3)*x/(1 - x/(x + 1/Q(k+1))).

G.f.: G(0)/2, where G(k) = 1 + 1/(1-x/(x+1/x/(2*k+1)/G(k+1))).

G.f.: (G(0) - 1)*x/(1+x) + 1, where G(k) = 1 + x*(2*k+1)/(1 - x/(x + 1/G(k+1))). (End)

For n even, a(n) = A001147(n/2) = A124794(3^(n/2)). a(n) is also the coefficient of x1*...*xn in Product_{1 <= i < j <= n} (1 + xi*xj). - Gus Wiseman, Dec 23 2018

a(n) = 2^(n/2)*Pochhammer(1/2, n/2)*(n+1 mod 2). - Peter Luschny, Jan 11 2023

Extensions

Edited by N. J. A. Sloane, Jan 06 2008

Better name by Sergei N. Gladkovskii, May 24 2013

Leading term 1 dropped, offset changed, and entry edited correspondingly by Andrey Zabolotskiy, Nov 07 2019